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A point mass and continuous collapse to a point mass in general relativity

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Abstract

An original way of presentation of the Schwarzschild black hole in the form of a point-like mass with making the use of the Dirac \(\delta \)-function, including a description of a continuous collapse to such a point mass, is given. A maximally generalized description restricted by physically reasonable requirements is developed. A so-called field-theoretical formulation of general relativity, being equivalent to the standard geometrical presentation of general relativity, is used. All of the dynamical fields, including the gravitational field, are considered as propagating in a background (curved or flat) spacetime. Namely these properties allow us to present a non-contradictive picture of the point mass description. The results can be useful for studying the structure of the black hole true singularities and could be developed for practical calculations in models with black holes.

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Notes

  1. It is a place to note that in [4, 5] it was clarified that the EF coordinates are the most preferable in their approach as well.

  2. Collapse from a finite radius has been suggested in [41] as well. However, we do not consider it here because principally it is the same, but formulae are significantly more complicated.

References

  1. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon Press, Oxford (1975)

    MATH  Google Scholar 

  2. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman and Company, San Francisco (1973)

    Google Scholar 

  3. Balasin, H., Nachbagauer, H.: Class. Quantum Grav. 10, 2271 . Preprint arXiv:gr-qc/9305009 (1993)

  4. Pantoja, N.R., Rago, H.: Int. J. Mod. Phys. D 11(9), 1479 . Preprint arXiv:gr-qc/0009053 (2002)

  5. Heinzle, J.M., Steinbauer, R.: J. Math Phys. 43(3), 1493 . Preprint arXiv:gr-qc/0112047) (2002)

  6. Abbott, B.P., et al.: (LIGO-Virgo Scientific Collaborations), Phys. Rev. Lett. 116(31 May 2016), 221101 . Preprint: arXiv:1602.03841 [gr-qc] (2016)

  7. Abbott, B.P., et al.: (LIGO-Virgo Scientific Collaborations), Phys. Rev. Lett. 116(15 June 2016), 241103 . Preprint: arXiv:1606.04855 [gr-qc] (2016)

  8. Abbott, B.P., et al.: (LIGO-Virgo Scientific Collaborations), Phys. Rev. X 6(4), 041015 . Preprint: arXiv:1606.04856 [gr-qc] (2016)

  9. Damour, T., Jaranowski, P., Schäfer, G.: Phys. Lett. B 513(1–2), 147 (2001)

    Article  ADS  Google Scholar 

  10. Schäfer, G.: Binary black holes and gravitational wave production: post-Newtonian analytic treatment. In: Fernández-Jambrina, L., González-Romero, L.M. (eds.) Current Trends in Relativistic Astrophysics. Lecture Notes in Physics, vol. 617. Springer, Berlin (2003)

  11. Schäfer, G.: Post-Newtonian methods: analytic results on the binary problem. In: Blanchet, L., Spallicci, A., Whiting, B. (eds.) Mass and motion in General Relativity, pp. 167–210. Springer, Dordrecht (2011)

  12. Blanchet, L.: Living Rev. Relativ. 17(2), 1. http://www.livingreviews.org/lrr-2014-2 (2014)

  13. Narlikar, J.V.: Random walk. In: Dadhich, N., Krishna Rao, J., Narlikar, J.V., Vishevara, C.V. (eds.) Relativity and Cosmology, pp. 171–183. Viley Eastern Limited, New Delhi (1985)

    Google Scholar 

  14. Grishchuk, L.P., Petrov, A.N., Popova, A.D.: Commun. Math. Phys. 94(3), 379 (1984)

    Article  ADS  Google Scholar 

  15. Popova, A., Petrov, A.: Int. J. Mod. Phys. A 3(11), 2651 (1988)

    Article  ADS  Google Scholar 

  16. Petrov, A.N.: Class. Quantum. Grav. 10(12), 2663 (1993)

    Article  ADS  Google Scholar 

  17. Petrov, A.N.: In: Christiansen, M.N., Rasmussen, T.K. (eds.) Classical and Quantum Gravity Research, pp. 79–160. Nova Science Publishers, New York. Preprint arXiv:0705.0019 [gr-qc] (2008)

  18. Petrov, A.N., Kopeikin, S.M., Lompay, R.R., Tekin, B.: Metric Theories of Gravity: Perturbations and Conservation Laws. De Gruyter, Germany (2017)

    Book  MATH  Google Scholar 

  19. Geroch, R., Traschen, J.: Phys. Rev. D 36(15 August), 1017 (1987)

  20. Colombeau, J.: J. Math. Anal. Appl. 94(1), 96 (1983)

    Article  MathSciNet  Google Scholar 

  21. Fiziev, P.: The gravitational field of massive non-charged point source in general relativity . Preprint arXiv:gr-qc/0412131) (2004)

  22. Goswami, R., Joshi, P.S., Vaz, C., L, W.: Phys. Rev. D 70(8), 084038 . Preprint arXiv:gr-qc/0410041 (2004)

  23. Cadoni, M., Magnemi, S.: Mod. Phys. Lett. A 20(38), 2919 . Preprint arXiv:gr-qc/0503059 (2005)

  24. Lundgren, A.P., Schmekel, B.S., York, J.W.: Phys. Rev. D 75(8), 084026 . Preprint arXiv:gr-qc/0610088 (2007)

  25. Brown, J.D., York, J.W.: Phys. Rev. D 47, 1407 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  26. Katanaev, M.O.: Gen. Relativ. Grav. 45(10), 1861 . Preprint arXiv:1207.3481 [gr-qc] (2013)

  27. Pitts, J.B., Schieve, W.C.: Found. Phys. 34(2), 211 . Preprint arXiv:gr-qc/0406102 (2004)

  28. Petrov, A.N., Narlikar, J.V.: Found. Phys. 26(9), 1201 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  29. Petrov, A.N.: Found. Phys. Lett. 18(5), 477 . Preprint arXiv:gr-qc/0503082 (2005)

  30. Deser, S.: Gen. Relativ. Grav. 1, 9 . Preprint arXiv:gr-qc/0411023(1970)

  31. Gelfand, I.M., Shilov, G.E.: Generalized Functions, Vol. 1. Properties and Operations. Academic Press, New York (1964)

    Google Scholar 

  32. Bizon, P., Malec, E.: Class. Quantum Grav. 3, L123 (1986)

    Article  ADS  Google Scholar 

  33. O’Murchadha, N.: J. Math. Phys. 27(8), 2111 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  34. Kennefick, D., O’Murchadha, N.: Class. Quantum. Grav. 12(1), 149 (1995)

    Article  ADS  Google Scholar 

  35. Soloviev, V.O.: Theor. Math. Phys 65, 1240 (1985)

    Article  Google Scholar 

  36. Petrov, A.N.: Int. J. Mod. Phys. D 4(4), 451 (1995)

    Article  ADS  Google Scholar 

  37. Petrov, A.N.: Int. J. Mod. Phys. D 6(2), 239 (1997)

    Article  ADS  Google Scholar 

  38. Eddington, A.S.: Nature 113(February 9), 192 (1924)

  39. Finkelstein, D.: Phys. Rev 110(4), 965 (1958)

    Article  ADS  Google Scholar 

  40. Oppenheimer, J.R., Snyder, H.: Phys. Rev. 56(September 1), 455 (1939)

  41. Kanai, Y., Siino, M., Hosoya, A.: Prog. Theor. Phys. 125(May 1), 1053 Preprint arXiv:1008.0470 [gr-qc] (2011)

  42. Peinlevé, P.: C. R. Acad. Sci. (Paris) 173(October 24), 677 (1921)

  43. Gullstrand, A.: Arkiv. Mat. Astron. Fys. 16(8), 1 (1922)

    Google Scholar 

  44. Hamilton, A.J.S., Lisle, J.P.: Am. J. Phys. 76(6), 519 . Preprint arXiv:gr-qc/0411060 (2008)

  45. Lin, C.Y., Soo, C.: Phys. Lett. B 671(4-5), 493 . Preprint arXiv:0810.2161 [gr-qc] (2009)

  46. Jan, X., Molina, A.: Preprint arXiv:gr-qc/0411060 (2016)

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  1. Sternberg Astronomical Institute, Moscow MV Lomonosov State University, Universitetskii pr. 13, Moscow, 119992, Russia

    Alexander N. Petrov

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  1. Alexander N. Petrov
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Petrov, A.N. A point mass and continuous collapse to a point mass in general relativity. Gen Relativ Gravit 50, 6 (2018). https://doi.org/10.1007/s10714-017-2326-4

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